Project #8 -- Quality Control


Introduction

In modern manufacturing plants, much of the work is done by robots. While robots are very efficient, they are less cognizant of errors than are human beings. Thus, in order to detect and respond rapidly to problems, quality control measures are incorporated into the production process at many stages.

For example, a company that makes pipeline needs to know that each end of a segment of pipe has been cut well. Generally, a segment of pipe has been cut well if each end of the segment is circular (there may be times when an elliptical opening is desired, but they are few and far between). For the purposes of this exercise, we define bad cuts as those which result in elliptical openings or, worse, openings that are not smooth. That is, the mouth of each pipe should be circular.


Mathematics Needed

Geometric ideas from linear algebra; dot product, cross product, norm


Assignment

Assume you are working in the quality control department of a company that makes pipeline, as described above. You are part of a team which has been assigned the task of developing a method for detecting errors in the cutting phase of production. Any such system must:
  1. take measurements on the mouth of a segment of pipe, and
  2. from those measurements, determine whether the mouth of the pipe is circular.
Your team has been delegated the latter of these steps. In practice, a computer will receive information about the location of ten points on the mouth of a pipe segment. Your goal is to use the concepts and techniques of linear algebra to develop an algorithm for determining, from such a data set, whether the mouth of the pipe is circular. To show that your method works, you will apply it to each of the data sets assigned to your team (found below). If one of the data sets describes a set of points that lie on a circle, you should determine the center and radius of that circle; if not, you should describe how you've made that determination.

You must submit to your superior, who understands the basics of linear algebra, a report detailing your methods, reasoning and conclusions.


The Report

You should think of this report as an unofficial, professional report of your findings (remember, it's to your boss, so it has to be clear, and concise). The memo should be divided into (at least) the following sections: introduction and summary, method, analysis, and testing.

Introduction and Summary : The goal of this section is a functional one. Your supervisors need to have a short summary of documents so they can quickly determine whether yours is the report they need or want. This section should give a brief account of the subject matter discussed in the document. Someone should be able to read this first section of your work and understand what the problem is, and your response to it. You should also provide a brief outline of your method; include only the major ideas, and describe them in generality.

Method: This is the heart of the report, and the most important part of this assignment. In this section you should clearly describe the algorithm you've devised. Be sure to clearly explain to your superiors the method(s) and ideas you used to devise ths algorithm. Your algorithm should be soundly based in valid ideas from linear algebra, and should offer a sensible solution to the problem.

Analysis: In this section you should think about what might go wrong with your method. For example, I suggest that you set Maple's accuracy to four digits (use the command Digits:=4;). What happens if you use the default accuracy of ten decimal places? and what if you used only one, or two decimal places? What benefits and/or detriments are incurred by greater or lesser accuracy? Are there other factors that might affect your ability to make the correct decision?

Testing: The end result of this project is an algorithm which can be used to determine whether a set of ten points lies on a circle. In this section, you will show that your method works by determining which of the data sets describes points on a circle.


Remarks

The amount of calculation involved in solving this problem is potentially great. However, you have Maple/Matlab/Mathematica at your disposal and should make use of it (there's no reason for you to do all the tedious number crunching by hand). The use of the program should be well documented in your report (where you used it, and for what). Be forewarned: you should develop a strategy for attacking the problem before you approach Maple (it's only a program, after all, and won't be able to give you hints). When working with Maple/Mathematica/Whatever, you should set its accuracy at four digits for starters (as mentioned above).


Data Sets

Data Set One (x,y,z) 
8.461   8.721  6.895
8.433   8.344  6.876
7.113  13.91   4.729
8.428  10.22   6.772
8.341   7.602  6.784
4.261   3.092  1.261
6.937  14.12   4.469
8.000   6.195  6.375
7.312   4.694  5.481
6.751  14.30   4.198

Data Set Two (x,y,z) 
6.983   5.002   6.136  
6.278   5.243   4.835
6.624   5.053   5.476
8.209   5.990   8.314
8.879   8.360   9.472
8.852   8.114   9.433
6.744   5.020   5.696
8.828  10.340   9.295
5.676   6.008   3.698
8.899   8.609   9.498

There will be no more data sets.

Points

This project will be worth 25 points towards your final grade. The point breakdown for this project is as follows:

Project adapted from the web pages of Carl V. Lutzer of University of Kentucky, who adapted it from Carl C. Cowen of Purdue University.